Summary: Problem Set 11
For this entire problem set, R = k[x1, x2, . . . , xn] with k a eld.
Let F k[x1, x2, . . . , xn]. F can be written as a sum F = i Fi where
each Fi is homogeneous of degree i. Recall that an ideal, I, is homogeneous if
F I = Fi I for each i.
Problem 1. Prove that an ideal is homogeneous the ideal is generated by
a nite set of homogeneous forms.
Problem 2. Let R = k[x, y, z]. Let F be an irreducible form of degree t. Let
= (V (F)). Consider the exact sequence 0 R
×F
R 0. Let
d = {Forms of degree d in }.
a) Show that d is a nite dimensional k-vector space.
b) Find dimk(d).
Recall that to Pn
we can associate a dual space Pn
. Each point in Pn
corre-
sponds to a hyperplane in Pn
by [a1 : a2 : · · · : an] V ( i aiyi). To a linear